5,366 research outputs found
k-Step Nilpotent Lie Algebras
The classification of complex of real finite dimensional Lie algebras which
are not semi simple is still in its early stages. For example the nilpotent Lie
algebras are classified only up to the dimension 7. Moreover, to recognize a
given Lie algebra in a classification list is not so easy. In this work we
propose a different approach to this problem. We determine families for some
fixed invariants, the classification follows by a deformation process or
contraction process. We focus on the case of 2 and 3-step nilpotent Lie
algebras. We describe in both cases a deformation cohomology of this type of
algebras and the algebras which are rigid regarding this cohomology. Other
-step nilpotent Lie algebras are obtained by contraction of the rigid ones.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1201.267
Characteristically nilpotent Lie algebras : a survey
We review the known results about characteristically nilpotent complex Lie
algebras, as well as we comment recent developements in the theory.Comment: Latex, 42 page
Breadth and characteristic sequence of nilpotent Lie algebras
The notion of breadth of a nilpotent Lie algebra was introduced by B.
Khuhirun, K.C. Misra and E. Stitzinger and used to approach problems of
classification up to isomorphism. In the present paper, we study this invariant
in terms of characteristic sequence, another invariant introduced by M. Goze
and J.M. Ancochea-Bermudez. This permits to complete the determination of Lie
algebras of breadth 2 and to begin the work for Lie algebras with breadth
greater than 2.Comment: 12 page
Enveloping algebras of restricted Lie superalgebras satisfying non-matrix polynomial identities
Let L be a restricted Lie superalgebra with its enveloping algebra u(L) over
a field F of characteristic p>2. A polynomial identity is called non-matrix if
it is not satisfied by the algebra of 2\times 2 matrices over F. We
characterize L when u(L) satisfies a non-matrix polynomial identity. In
particular, we characterize L when u(L) is Lie solvable, Lie nilpotent, or Lie
super-nilpotent.Comment: proofs of some of the statements are shortened along with transparent
ideas, some typos are fixe
Lie algebras with associative structures. Applications to the study of 2-step nilpotent Lie algebras
We investigate Lie algebras whose Lie bracket is also an associative or cubic
associative multiplication to characterize the class of nilpotent Lie algebras
with a nilindex equal to 2 or 3. In particular we study the class of 2-step
nilpotent Lie algebras, their deformations and we compute the cohomology which
parametrize the deformations in this class.Comment: 17 page
The Nash-Moser Theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras
We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the
context of deformations of Lie algebras and we discuss some aspects of the
scope of this theorem in connection with the polynomial ideal associated to the
variety of nilpotent Lie algebras. This allows us to introduce the space
, and certain subspaces of it, that
provide fine information about the deformations of in the
variety of -step nilpotent Lie algebras.
Then we focus on degenerations and rigidity in the variety of -step
nilpotent Lie algebras of dimension with and, in particular, we
obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent
Lie algebras of dimension 7. We also recover some known results and point out a
possible error in a published article related to this subject.Comment: Accepted in J. of Pure and Applied Algebra. The structure of the
paper has been bearly modified to follow the referee's suggestion
The geometric classification of Leibniz algebras
We describe all rigid algebras and all irreducible components in the variety
of four dimensional Leibniz algebras over In
particular, we prove that the Grunewald--O'Halloran conjecture is not valid and
the Vergne conjecture is valid for $\mathfrak{Leib}_4.
Minimal metrics on nilmanifolds
A left invariant metric on a nilpotent Lie group is called minimal, if it
minimizes the norm of the Ricci tensor among all left invariant metrics with
the same scalar curvature. Such metrics are unique up to isometry and scaling
and the groups admitting a minimal metric are precisely the nilradicals of
(standard) Einstein solvmanifolds. If is endowed with an invariant
symplectic, complex or hypercomplex structure, then minimal compatible metrics
are also unique up to isometry and scaling. The aim of this paper is to give
more evidence of the existence of minimal metrics, by presenting several
explicit examples. This also provides many continuous families of symplectic,
complex and hypercomplex nilpotent Lie groups. A list of all known examples of
Einstein solvmanifolds is also given.Comment: 18 page
Quantizations of regular functions on nilpotent orbits
We study the quantizations of the algebras of regular functions on nilpotent
orbits. We show that such a quantization always exists and is unique if the
orbit is birationally rigid. Further we show that, for special birationally
rigid orbits, the quantization has integral central character in all cases but
four (one orbit in E_7 and three orbits in E_8). We use this to complete the
computation of Goldie ranks for primitive ideals with integral central
character for all special nilpotent orbits but one (in E_8). Our main
ingredient is results on the geometry of normalizations of the closures of
nilpotent orbits by Fu and Namikawa.Comment: 17 page
Generic properties of 2-step nilpotent Lie algebras and torsion-free groups
To define the notion of a generic property of finite dimensional 2-step
nilpotent Lie algebras we use standard correspondence between such Lie algebras
and points of an appropriate algebraic variety, where a negligible set is one
contained in a proper Zariski-closed subset. We compute the maximal dimension
of an abelian subalgebra of a generic Lie algebra and give a sufficient
condition for a generic Lie algebra to admit no surjective homomorphism onto a
non-abelian Lie algebra of a given dimension. Also we consider analogous
questions for finitely generated torsion free nilpotent groups of class 2.Comment: 16 page
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